Soak the slices of bread in the cup of olive oil. In a large pot, heat remaining olive oil and gently sauté onion until translucent. Stir in tomatoes and red pepper. In a separate pot, bring mineral water to a boil, then pour water into the pot with the vegetables and continue to boil for at least 10 more minutes. Season soup to taste with salt and pepper. Place into each bowl 2 slices of soaked bread, then pour the soup over it. Sprinkle with Pecorino. Soup can be served either hot or cold.